**16.3** The Fundamental

### Theorem for Line Integrals

### The Fundamental Theorem for Line Integrals

We know that Part 2 of the Fundamental Theorem of Calculus can be written as

*where F′ is continuous on [a, b].*

We also called Equation 1 the Net Change Theorem:

The integral of a rate of change is the net change.

### The Fundamental Theorem for Line Integrals

If we think of the gradient vector ∇f of a function f of two or
*three variables as a sort of derivative of f, then the following *
theorem can be regarded as a version of the Fundamental
Theorem for line integrals.

### Example 1

Find the work done by the gravitational field

*in moving a particle with mass m from the point (3, 4, 12) to *
*the point (2, 2, 0) along a piecewise-smooth curve C.*

Solution:

**We know that F is a conservative vector field and, in fact,**
**F = ∇f, where**

*Example 1 – Solution*

Therefore, by Theorem 2, the work done is
*W = *

### ∫

_{C}**F**

**dr =**### ∫

_{C}*∇f*

**dr***= f(2, 2, 0) – f(3, 4, 12) *

cont’d

### Independence of Path

### Independence of Path

*Suppose C*_{1} *and C*_{2} are two piecewise-smooth curves

**(which are called paths) that have the same initial point A***and terminal point B.*

We know that, in general,

### ∫

_{C}_{1}

**F**

*≠*

**dr**### ∫

_{C}_{2}

**F**

*implication of Theorem 2 is that*

**dr. But one**### ∫

_{C}_{1}

*∇f*

**dr =**### ∫

_{C}_{2}

*∇f*

**dr**whenever ∇f is continuous. In other words, the line integral
*of a conservative vector field depends only on the initial *

point and terminal point of a curve.

### Independence of Path

* In general, if F is a continuous vector field with domain D,*
we say that the line integral

### ∫

_{C}

^{F }^{}

**dr is independent of****path if **

### ∫

_{C}_{1}

^{F}^{}

^{dr = }### ∫

_{C}_{2}

^{F}^{}

**dr for any two paths C**_{1}

*and C*

_{2}

*in D*that have the same initial points and the same terminal

points.

*With this terminology we can say that line integrals of *
*conservative vector fields are independent of path.*

**A curve is called closed if its**
terminal point coincides with
its initial point, that is,

**r(b) = r(a). (See Figure 2.)****Figure 2**

A closed curve

### Independence of Path

If

### ∫

_{C}**F**

**dr is independent of path in D and C is any closed***path in D, we can choose any two points A and B on C and*

*regard C as being composed of the path C*

_{1}

*from A to B*

*followed by the path C*_{2} *from B to A. (See Figure 3.)*

**Figure 3**

### Independence of Path

Then

### ∫

_{C}**F**

**dr =**### ∫

_{C}_{1}

**F**

**dr +**### ∫

_{C}_{2}

**F**

**dr =**### ∫

_{C}_{1}

**F**

**dr –**### ∫

_{–C}_{2}

**F**

**dr = 0***since C*_{1} *and –C*_{2} have the same initial and terminal points.

Conversely, if it is true that

### ∫

_{C}

^{F }^{}

**dr = 0 whenever C is a***closed path in D, then we demonstrate independence of*path as follows.

*Take any two paths C*_{1} *and C*_{2} *from A to B in D and define *
*C to be the curve consisting of C*_{1} *followed by –C*_{2}.

### Independence of Path

Then

0 =

### ∫

_{C}

^{F }^{}

^{dr = }### ∫

_{C}_{1}

^{F }^{}

^{dr + }### ∫

_{–C}_{2}

^{F }^{}

^{dr = }### ∫

_{C}_{1}

^{F }^{}

^{dr –}### ∫

_{C}_{2}

^{F }^{}

^{dr}and so

### ∫

_{C}_{1}

^{F}^{}

^{dr = }### ∫

_{C}_{2}

^{F}^{}

^{dr.}Thus we have proved the following theorem.

Since we know that the line integral of any conservative
**vector field F is independent of path, it follows that**

### Independence of Path

The physical interpretation is that the work done by a conservative force field as it moves an object around a closed path is 0.

*The following theorem says that the only vector fields that *
are independent of path are conservative. It is stated and
proved for plane curves, but there is a similar version for
space curves.

### Independence of Path

**We assume that D is open, which means that for every **

*point P in D there is a disk with center P that lies entirely in *
*D. (So D doesn’t contain any of its boundary points.)*

**In addition, we assume that D is connected: this means ***that any two points in D can be joined by a path that lies *
*in D.*

The question remains: how is it possible to determine

**whether or not a vector field F is conservative? Suppose it **
* is known that F = P i + Q j is conservative, where P and Q *
have continuous first-order partial derivatives. Then there is

### Independence of Path

Therefore, by Clairaut’s Theorem,

The converse of Theorem 5 is true only for a special type of region.

### Independence of Path

**To explain this, we first need the concept of a simple **
**curve, which is a curve that doesn’t intersect itself **

**anywhere between its endpoints. [See Figure 6; r (a) = r (b) ****for a simple closed curve, but r(t**_{1}**) ≠ r(t**_{2}) when

*a < t*_{1 }*< t*_{2 }*< b.]*

### Independence of Path

In Theorem 4 we needed an open connected region. For the next theorem we need a stronger condition.

**A simply-connected region in the plane is a connected**
*region D such that every simple closed curve in D*

*encloses only points that are in D.*

Notice from Figure 7 that, intuitively speaking, a

simply-connected region contains no hole and can’t

consist of two separate pieces.

**Figure 7**

### Independence of Path

In terms of simply-connected regions, we can now state a partial converse to Theorem 5 that gives a convenient

method for verifying that a vector field on is conservative.

### Example 2

Determine whether or not the vector field

* F (x, y) = (x – y) i + (x – 2) j*
is conservative.

Solution:

*Let P (x, y) = x – y and Q(x, y) = x – 2. Then*

Since ∂P/∂y **≠ ∂Q/∂x, F is not conservative by Theorem 5.**

### Conservation of Energy

### Conservation of Energy

Let’s apply the ideas of this chapter to a continuous force
**field F that moves an object along a path C given by r(t),***a ≤ t ≤ b, where r(a) = A is the initial point and r(b) = B is *
*the terminal point of C.*

According to Newton’s Second Law of Motion, the force
**F(r(t)) at a point on C is related to the acceleration **

**a(t) = r**″ (t) by the equation

**F(r(t)) = m r**″(t)

So the work done by the force on the object is

### Conservation of Energy

Therefore

**where v = r**′ is the velocity.

(Fundamental Theorem of Calculus)

**(By formula d/dt [u(t)**** v(t)] **

**= u′(t) v(t) + u(t) v′(t)) **

### Conservation of Energy

The quantity that is, half the mass times the
**square of the speed, is called the kinetic energy of the **
object. Therefore we can rewrite Equation 15 as

*W = K(B) – K(A)*

*which says that the work done by the force field along C is *
*equal to the change in kinetic energy at the endpoints of C.*

**Now let’s further assume that F is a conservative force field; **

**that is, we can write F = ∇f.**

### Conservation of Energy

**In physics, the potential energy of an object at the **
*point (x, y, z) is defined as P(x, y, z) = –f(x, y, z), so we *
**have F = –∇P.**

Then by Theorem 2 we have

W =

### ∫

_{C}

^{F }^{}

^{dr = –}### ∫

_{C}*∇P*

^{}

**dr = –[P(r(b)) – P(r(a))]***= P(A) – P(B)*

### Conservation of Energy

Comparing this equation with Equation 16, we see that
*P(A) + K(A) = P(B) + K(B)*

*which says that if an object moves from one point A to *

*another point B under the influence of a conservative force *
field, then the sum of its potential energy and its kinetic

energy remains constant.

**This is called the Law of Conservation of Energy and it is **
*the reason the vector field is called conservative.*